Classifications

• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
  • G01Q10/00—Scanning or positioning arrangements, i.e. arrangements for actively controlling the movement or position of the probe
• • B—PERFORMING OPERATIONS; TRANSPORTING
  • B23—MACHINE TOOLS; METAL-WORKING NOT OTHERWISE PROVIDED FOR
  • B23Q—DETAILS, COMPONENTS, OR ACCESSORIES FOR MACHINE TOOLS, e.g. ARRANGEMENTS FOR COPYING OR CONTROLLING; MACHINE TOOLS IN GENERAL CHARACTERISED BY THE CONSTRUCTION OF PARTICULAR DETAILS OR COMPONENTS; COMBINATIONS OR ASSOCIATIONS OF METAL-WORKING MACHINES, NOT DIRECTED TO A PARTICULAR RESULT
  • B23Q15/00—Automatic control or regulation of feed movement, cutting velocity or position of tool or work
• • G—PHYSICS
  • G01—MEASURING; TESTING
  • G01Q—SCANNING-PROBE TECHNIQUES OR APPARATUS; APPLICATIONS OF SCANNING-PROBE TECHNIQUES, e.g. SCANNING PROBE MICROSCOPY [SPM]
  • G01Q10/00—Scanning or positioning arrangements, i.e. arrangements for actively controlling the movement or position of the probe
  • G01Q10/04—Fine scanning or positioning
  • G01Q10/06—Circuits or algorithms therefor
  • G01Q10/065—Feedback mechanisms, i.e. wherein the signal for driving the probe is modified by a signal coming from the probe itself
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
  • G03F7/20—Exposure; Apparatus therefor
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F7/00—Photomechanical, e.g. photolithographic, production of textured or patterned surfaces, e.g. printing surfaces; Materials therefor, e.g. comprising photoresists; Apparatus specially adapted therefor
  • G03F7/70—Microphotolithographic exposure; Apparatus therefor
  • G03F7/70691—Handling of masks or workpieces
  • G03F7/70716—Stages
  • G03F7/70725—Stages control
• • G—PHYSICS
  • G03—PHOTOGRAPHY; CINEMATOGRAPHY; ANALOGOUS TECHNIQUES USING WAVES OTHER THAN OPTICAL WAVES; ELECTROGRAPHY; HOLOGRAPHY
  • G03F—PHOTOMECHANICAL PRODUCTION OF TEXTURED OR PATTERNED SURFACES, e.g. FOR PRINTING, FOR PROCESSING OF SEMICONDUCTOR DEVICES; MATERIALS THEREFOR; ORIGINALS THEREFOR; APPARATUS SPECIALLY ADAPTED THEREFOR
  • G03F9/00—Registration or positioning of originals, masks, frames, photographic sheets or textured or patterned surfaces, e.g. automatically
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B19/00—Programme-control systems
  • G05B19/02—Programme-control systems electric
  • G05B19/18—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form
  • G05B19/19—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path
  • G05B19/21—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path using an incremental digital measuring device
  • G05B19/23—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path using an incremental digital measuring device for point-to-point control
  • G05B19/231—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path using an incremental digital measuring device for point-to-point control the positional error is used to control continuously the servomotor according to its magnitude
  • G05B19/235—Numerical control [NC], i.e. automatically operating machines, in particular machine tools, e.g. in a manufacturing environment, so as to execute positioning, movement or co-ordinated operations by means of programme data in numerical form characterised by positioning or contouring control systems, e.g. to control position from one programmed point to another or to control movement along a programmed continuous path using an incremental digital measuring device for point-to-point control the positional error is used to control continuously the servomotor according to its magnitude with force or acceleration feedback only
• • G—PHYSICS
  • G12—INSTRUMENT DETAILS
  • G12B—CONSTRUCTIONAL DETAILS OF INSTRUMENTS, OR COMPARABLE DETAILS OF OTHER APPARATUS, NOT OTHERWISE PROVIDED FOR
  • G12B5/00—Adjusting position or attitude, e.g. level, of instruments or other apparatus, or of parts thereof; Compensating for the effects of tilting or acceleration, e.g. for optical apparatus
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B2219/00—Program-control systems
  • G05B2219/30—Nc systems
  • G05B2219/49—Nc machine tool, till multiple
  • G05B2219/49279—Nanometric xy table
• • G—PHYSICS
  • G05—CONTROLLING; REGULATING
  • G05B—CONTROL OR REGULATING SYSTEMS IN GENERAL; FUNCTIONAL ELEMENTS OF SUCH SYSTEMS; MONITORING OR TESTING ARRANGEMENTS FOR SUCH SYSTEMS OR ELEMENTS
  • G05B2219/00—Program-control systems
  • G05B2219/30—Nc systems
  • G05B2219/49—Nc machine tool, till multiple
  • G05B2219/49281—X y table positioned by vibration

Definitions

• the present invention relates to a positioning method and system and in particular a method and system for nanopositioning apparatuses.
• the invention has been developed primarily for use in a nanopositioning system or apparatus, and will be described hereinafter with reference to this application. However, it will be appreciated that the invention is not limited to this particular field of use. In particular, it is contemplated that the invention is also applicable to positioning systems, apparatuses and methods where precise and accurate displacements of an object mounted in the positioning system or to a positioning apparatus are required.
• Nanopositioning systems and apparatuses are used to generate fine mechanical displacements with resolution frequently down to the atomic scale. Such systems and apparatuses include fiber aligners, beam scanners, and lateral positioning platforms. Other applications of nanopositioning apparatuses in nanotechnology include scanning probe microscopy (SPM), scanning tunnelling microscopy (STM), atomic force microscopy (AFM), nanofabrication systems, precision machining, optical switching and cell physiology research.
• SPM scanning probe microscopy
• STM scanning tunnelling microscopy
• AFM atomic force microscopy
• nanofabrication systems precision machining, optical switching and cell physiology research.
• piezoelectric actuators are universally employed in nanopositioning applications to provide the greatest possible positioning accuracy, which is also known as tracking performance.
• the positioning accuracy of piezoelectric actuators is severely limited by hysteresis over relatively large displacements and creep at low frequencies.
• Hysteresis occurs where the platform position becomes a function of the past history of its movement. This is due to the piezoelectric response to an input voltage being a function of the previous voltage history. Creep occurs when the platform slowly drifts in the direction of recent movements. These slow drifts in position occur due to previous input voltages applied to the nanopositioning apparatus.
• all nanopositioning systems typically require some form of feedback or feedforward control to reduce non-linearity caused by hysteresis and creep.
• a first aspect of the invention provides a system for positioning an object, comprising: a fixed base; a support for the object; an actuator for applying a force to displace the support relative to the fixed base; a sensor for measuring the load force on the support, and a controller for processing the measured load force to control the position of the support and/or damp at least one resonance frequency of the system.
• the controller processes the measured load force in a feedback loop.
• the controller adjusts the position of the support in response to the measured load force.
• the controller controls the actuator to adjust the position of the support.
• the controller calculates the displacement of the support from the measured load force.
• the displacement of the support is calculated by the relationship: where d is the displacement of the support;
• F is the measured load force
• M p is the mass of the support
• s is the Laplace transform parameter
• Cf is the flexure damping ratio
• kf is the flexure stiffness.
• the force sensor generates an output voltage corresponding to the measured load force. More preferably, the displacement of the support is calculated as a proportion of the output voltage of the force sensor. In one preferred form, the displacement of the support is calculated by the relationship:
• V s is the output voltage of the measured load force
• F is the measured load force
• g s is the force sensor gain
• Cf is the flexure damping ratio; and &/is the flexure stiffness.
• the force sensor is calibrated using the charge and/or the voltage of the force sensor.
• the controller processes the measured load force in the feedback loop at frequencies above a predetermined cross-over frequency ⁇ c .
• the cross-over frequency is above a cut-off frequency of the force sensor.
• the cross-over frequency is determined according to the relationship:
• the system comprises a position sensor for measuring the position of the support. More preferably, the measured support position is used to calculate the displacement of the support.
• the controller calculates the displacement of the support from the input voltage of the actuator and the open-loop response of the system. In either case, it is preferred that the controller processes the calculated displacement in the feedback loop at frequencies below the cross-over frequency ⁇ c .
• the system comprises a displacement sensor for measuring the displacement of the support. More preferably, the controller processes the measured displacement in the feedback loop at frequencies below the cross-over frequency ⁇ c . Preferably, the controller processes the measured load force in the feedback loop to increase the damping ratio of the system.
• the at least one resonance frequency is the first resonance mode of the system.
• the system comprises multiple resonance modes and the controller damps one or more resonance modes of the system.
• the force sensor is interposed at least partially between the support and the actuator.
• the force sensor is a piezoelectric transducer.
• the controller adds a feedforward input to the feedback loop to improve the closed-loop response of the system.
• a second aspect of the invention provides a method of controlling a system for positioning an object, the system comprising a fixed base, a support for the object and an actuator for applying a force to the support, the method comprising the steps of: actuating the actuator to apply the force to displace the support relative to the fixed base; measuring the load force on the support, and processing the measured load force to control the position of the support and/or damp at least one resonance frequency of the system.
• the processing step comprises processing the measured load force in a feedback loop.
• the method comprises the step of adjusting the position of the support in response to the measured load force.
• the method comprises the step of controlling the actuator to adjust the position of the support.
• the method comprises the step of calculating the displacement of the support from the measured load force.
• the displacement of the support is calculated by the relationship: where d is the displacement of the support;
• F is the measured load force
• Mp is the mass of the support
• s is the Laplace transform parameter
• c / is the flexure damping ratio
• kf is the flexure stiffness.
• the load force measuring step comprises using a force sensor to measure the load force. More preferably, the force sensor generates an output voltage corresponding to the measured load force. Even more preferably, the method comprises the step of calculating the displacement of the support as a proportion of the output voltage of the force sensor. In one preferred form, the displacement of the support is calculated by the relationship:
• V s is the output voltage of the measured load force
• F is the measured load force
• g s is the force sensor gain
• Mp is the mass of the support
• ⁇ is the Laplace transform parameter
• Cf is the flexure damping ratio
• kf is the flexure stiffness
• the method comprises the step of calibrating the force sensor using the charge and/or the voltage of the force sensor.
• the processing step comprises processing the measured load force in the feedback loop at frequencies above the cross-over frequency ⁇ c .
• the cross-over frequency ⁇ c is above a cut-off frequency of the force sensor.
• the cross-over frequency ⁇ c is determined according to the relationship:
• Ri n is the voltage buffer input impedance
• C is the capacitance of the force sensor.
• the method comprises the steps of measuring the position of the support and calculating the displacement of the support from the measured support position.
• the method comprises the step of calculating the displacement of the support from the input voltage of the actuator and the open-loop response of the system.
• the method further comprises the step of processing the calculated displacement in the feedback loop at frequencies below the cross-over frequency ⁇ c .
• the method comprises the step of measuring the displacement of the support. More preferably, the method comprises the step of processing the measured displacement in the feedback loop processes at frequencies below a predetermined crossover frequency ⁇ c .
• the method comprises the step of processing the measured load force in the feedback loop to increase the damping ratio of the system.
• the at least one resonance frequency is the first resonance mode of the system.
• the system comprises multiple resonance modes and the controller damps one or more of the resonance modes.
• the method comprises the step of adding a feedforward input to the feedback loop to improve the closed-loop response of the system.
• the system is a nanopositioning system. That is, the system enables the precise motion and positioning of the object on the nano-scale.
• measurement of actuator load force is used as a feedback variable for both tracking (position) and/or damping control in the feedback loop.
• the use of the measured load force in the feedback control loop results in a zero-pole ordering. This allows a simple integral controller to provide excellent tracking and damping performance without any limitations on the gain.
• the system is thus stable with a theoretically infinite gain margin and 90 degrees phase margin. Positioning noise is also substantially reduced, as a piezoelectric force sensor generates less noise than a capacitive or inductive position sensor.
• Figure 1 is a schematic drawing of a circuit representing a typical nanopositioning system G
• Figure 2a are frequency response graphs of the nanopositioning system of Figure 1;
• Figure 2b are loop-gain graphs of the nanopositioning system of Figure 1;
• Figure 3 is a schematic drawing of a single degree of freedom nanopositioning stage in accordance with one embodiment of the invention;
• Figure 4 is a schematic drawing of a method in accordance with one embodiment of the invention for implementation in the system of Figure 3;
• Figure 5a is a schematic drawing of a circuit representing a nanopositioning system according to another embodiment of the invention.
• Figure 5b is a closed-loop frequency response graph for the system of Figure 5a
• Figure 5c is a position noise graph for the system of Figure 5a
• Figure 6a is a drawing of a Noliac monolithic stack actuator for use in the nanopositioning stage of Figure 3;
• Figure 6b is a schematic drawing representing the actuator of Figure 4.
• Figure 7 is a schematic drawing of an electrical model of a force sensor and associated charge measurement circuit for use in the system of Figure 3;
• Figure 8 is a schematic drawing of a mechanical circuit representing the system of Figure 3;
• Figure 9 are frequency response graphs of the force transfer function for the system of Figure 3;
• Figure 10 is a schematic drawing of a circuit representing a nanopositioning system in accordance with another embodiment of the invention.
• Figure 11 is a root-locus of the closed-loop poles for the integral controller C from Figure 10;
• Figure 12 are frequency response graphs comparing the open-loop and closed-loop frequency responses of the system of Figure 9;
• Figure 13 is a schematic drawing of a single degree of freedom nanopositioning stage in accordance with a further embodiment of the invention.
• Figure 14a is a schematic drawing of a circuit representing the nanopositioning system of Figure 13;
• Figure 14b is a closed-loop frequency response graph for the system of Figures 13 and 14a;
• Figure 14c is a position noise graph for the system of Figures 13 and 14a;
• Figure 15 is a schematic drawing of a method in accordance with another embodiment of the invention for implementation in the system of Figure 13;
• Figure 16a is a schematic drawing of a circuit representing a nanopositioning system according to another embodiment of the invention;
• Figure 16b is a closed-loop frequency response graph for the system of Figure 16a;
• Figure 16c is a position noise graph for the system of Figures 16a;
• Figures 17 and 18 are frequency response graphs comparing the performance of the embodiments of Figures 5 and 13 to 16 with a basic integral controller;
• Figures 19a, 19b and 19c are drawings of a nanopositioning platform and two types of associated force sensor, respectively, in accordance with an example of the invention.
• Figures 20a and 20b are the open-loop and closed-loop frequency response graphs illustrating the performance of the nanopositioning platform of Figure 19. PREFERRED EMBODIMENTS OF THE INVENTION
• a preferred application of the invention is in the field of nanopositioning systems, apparatuses and methods.
• Such systems and apparatuses typically have a fixed base with a movable platform or stage, with guiding flexures and/or mechanical linkages so as to constrain the movement of the platform or stage to a single degree of freedom.
• the object is mounted to or located on the platform or stage and the apparatus moves the platform to precisely position the object. Additional flexures and/or mechanical linkages may also be connected to the platform so that the platform can have multiple degrees of freedom, and can be constrained to move in a single degree of freedom at any one time.
• nanopositioning systems suffer from limitations due to hysteresis, creep and mechanical resonance.
• mechanical resonance limits the operation of the nanopositioning systems in terms of its positioning accuracy, speed and stability.
• G(s) 2 ⁇ ⁇ ⁇ ... (1) s 2 + 2 ⁇ r ⁇ s + ⁇ ; where ⁇ ,- is the resonance frequency; ⁇ is the damping ratio; and s is the Laplace transform parameter.
• the response of nanopositioning systems and apparatuses is improved by using a sensor and feedback loop.
• One common technique in commercial nanopositioning systems is sensor-based feedback using integral or proportional-integral control.
• Such controllers are simple, robust to modeling error, and due to high loop-gain at low- frequencies, effectively reduce piezoelectric non-linearity.
• the feedback gain ⁇ is also the closed-loop bandwidth ⁇ c/ (in radians per second).
• the maximum closed-loop bandwidth is equal to twice the product of the damping ratio ⁇ and the resonance frequency ⁇ r .
• the damping ratio ⁇ is usually in the order of 0.01, meaning that the maximum closed-loop bandwidth ⁇ c/ is less than 2% of the resonance frequency ⁇ r .
• Damping control uses a feedback loop to artificially increase the damping ratio ⁇ of a system. Due to equation (2), an increase in the damping ratio ⁇ allows a proportional increase in the feedback gain and closed-loop bandwidth, thus overcoming the bandwidth limitations of mechanical resonance.
• damping controllers alone cannot increase the closed-loop bandwidth too far beyond the resonance frequency, they have the advantage of being insensitive to variations in resonance frequency.
• damping controllers suppress, rather than invert, the mechanical resonance, they provide better rejection of external disturbances than inversion based systems.
• Damping control is frequently combined with tracking control to improve performance of the system.
• the tracking controller gain is still limited by stability margins and positioning resolution is dominated by sensor-induced noise.
• RMS root-mean-square
• the positioning noise is 0.2 nm RMS, or approximately 1.2 nm peak-to-peak, if the noise is normally distributed.
• the closed-loop bandwidth must be reduced to below 1 Hz, which is a severe limitation.
• high sensor-induced noise places a penalty on positioning resolution as bandwidth is increased.
• one embodiment of the invention provides a system for positioning an object in the form of a nanopositioning stage 1 having a fixed base 2 and a support in the form of a platform 3 for the object (not shown).
• a piezoelectric actuator 4 is mounted between the fixed base 2 and the platform 3, and applies a force to displace the platform 3 relative to the fixed base 2.
• a sensor in the form of a piezoelectric transducer 5 for measuring the load force on the platform 3 is interposed between the platform and the actuator 4.
• a controller 6 is electrically connected to the actuator 4 and the piezoelectric transducer 5 via connection lines 7 and 8, respectively, and processes the measured load force to control the position of the support (and thus the object) or damp at least one resonance frequency of the system 1.
• the stage 1 additionally has a plurality of flexures 9 for guiding displacement of the platform 3 so that movement of the platform is only in the direction and distance d, whilst constraining platform movement in the remaining degrees of freedom. That is, the flexures 9 ensure that the platform 3 moves slides or translates along a distance d in a single direction (ie. one degree of freedom). Mechanical linkages may be used in combination with or substitution for the flexures 9.
• the piezoelectric actuator 4 is actuated to apply a force to the platform 3 to displace it relative to the fixed base at step 11.
• the piezoelectric transducer measures the load force on the platform 3 at step 12.
• the measured load force is processed to control the position of the platform 3 or damp at least one resonance frequency of the system.
• the invention permits the use of a simple integral controller to provide tracking and damping performance of the nanopositioning system and method without limitations on the gain.
• an input signal corresponding to a desired position of the object on the platform 3 is sent to the controller 6, which converts it into a signal corresponding to the necessary displacement dj.
• the controller 6 then actuates the piezoelectric actuator 4 through the connection line 7.
• the actuator 4 then applies a force to the platform 3 to displace it a distance d.
• the piezoelectric transducer 5 measures the load force on the platform 3 and then sends a signal corresponding to the measured load force back to the controller 6 through the connection line 8.
• the measured load force signal is fed into a feedback loop of the controller 6 to calculate the actual displacement d, 2 in accordance with a predetermined proportional relationship, which will be described in more detail below.
• the controller 6 sends a signal through the connection line 7 to the piezoelectric actuator 4 to decrease or increase the force applied to the platform 3, and hence the load force on the platform 3, thus adjusting the position of the platform 3.
• the stage 1 is able to advantageously dynamically correct the platform position in response to the load force without being adversely affected by sensor-induced noise. This is because the piezoelectric transducer 5 generates much less noise than standard position sensors, such as inductive and capacitive position sensors.
• the controller 6 processes the measured load force in the feedback loop to increase the damping ratio ⁇ for the stage 1.
• the controller is able to damp the resonance frequency (or frequencies where there are multiple resonance modes) of the stage 1 dynamically in response to variations in the object mass, and thus does not require calibration of the stage 1 each time an object of differing mass is mounted to the platform 3.
• the controller 6 comprises a damping controller Cd and a tracking controller C t , separate from each other, and uses the measured load force to only damp the resonance frequency or frequencies of the system G.
• the tracking controller C 1 uses a displacement feedback loop to apply direct tracking of the stage 1.
• the displacement d must be obtained with a physical displacement sensor such as a capacitive, inductive or optical sensor.
• the low bandwidth of the integral tracking controller C t is significantly improved by adding the internal force feedback loop of the damping controller C d , as shown in Figure 5a.
• the damping controller C d eliminates the lightly damped resonance, gain- margin is drastically increased, allowing a proportional increase in tracking bandwidth.
• To find the closed-loop transfer function it is first convenient to find the transfer function of the internal loop. That is, the transfer function J du from u to d is
• Gdva is the transfer function from the applied voltage to displacement
• Cd is the damping controller using force feedback in accordance with the embodiment of the invention.
• Gvsva is the transfer function from the applied voltage to the sensed (output) voltage.
• the inventor has also discovered that using the measured load force as a feedback variable confers advantages over existing feedback loops, especially displacement feedback loops where a displacement measurement is used as a feedback variable.
• the system 1 and method 10 adequately represents the dominant dynamics exhibited by many nanopositioning geometries.
• a typical multi-layer monolithic Noliac stack actuator 25 comprising several layers 26 is used to represent the piezoelectric actuator 4, as best illustrated in Figure 6a.
• a schematic drawing of a circuit representing the stack actuator 25 is illustrated in Figure 6b, where the circuit includes the voltage dependent developed force F a , stiffness k a , effective mass M a and damping coefficient c a .
• the piezoelectric stack actuator 4 experiences an internal stress in response to an applied voltage.
• the voltage dependent developed force Fa represents this stress and is related to the "free" displacement d of the platform 3 by:
• ⁇ L is the change in actuator length (m)
• k a is the actuator stiffness (N/m).
• the developed force F a is most easily related to applied voltage by beginning with the standard expression for unrestrained linear stack actuators, where d 33 is the piezoelectric strain constant (m/V), n is the number of layers; and V a is the applied voltage. Combining equations (7) and (8) yields an expression for the developed force as a function of applied voltage:
• the developed force F a applied to the platform 3 is not the same as the load force on the platform 3 to effect displacement d, as some of the force F a is dissipated due to the combination of the stiffness and damping effects of the flexures and actuators, and the masses of the platform and actuator.
• the piezoelectric transducer 5 is used to minimise the additional mass and compliance associated with the force sensor.
• piezoelectric transducers advantageously provide high sensitivity and bandwidth with low-noise at high frequencies.
• the transducer 5 takes the form of a single wafer of piezoelectric material interposed or sandwiched between the platform 3 and the actuator 4.
• V - q nd ⁇ F ( ⁇ 4) s c s c s
• V s is the output voltage of the circuit
• q is the charge
• Cs is the output capacitance of the circuit
• n is the number of layers of the actuator
• d ⁇ 3 is the piezoelectric stain constant (m/V)
• F is the load force.
• Piezoelectric force sensors can also be calibrated using voltage rather than charge measurement.
• the generated charge is deposited on the internal capacitance of the transducer.
• the dynamics of the force sensor 5 are slightly altered. In effect, the transducer 5 is marginally stiffened.
• the open-circuit voltage of the piezoelectric force sensor is approximately: where V s is the output or measured voltage, n is the number of layers of the piezoelectric force sensor; da is the piezoelectric stain constant (m/V); F is the load force; and C is the transducer capacitance. Therefore, the scaling factor between the load force F and the measured (output) voltage Vs is — — Volts per Newton.
• the load force F can be calculated directly
• the transducer 5 senses the load force F upon the platform 3 (due to the action of the actuator 4) and converts the load force F into an output voltage V s .
• the transducer 5 transmits this output voltage V s as a signal through the connection line 8 to the controller 6, which then processes the output voltage signal in the feedback loop to control the position of the platform 3 and/or damp at least one resonant frequency of the stage 1, in this case the first resonance mode.
• the above proportional relationship enables the load force F to be readily measured in the system 1 , thus providing a convenient and suitable feedback variable to control the position of the platform 3 and/or damp the resonant frequencies of the system 1.
• this constant will be denoted as g ⁇ so that:
• Figure 3 is shown in Figure 8.
• the developed actuator force F n results in a load force F and a platform displacement d.
• the stiffness and damping coefficients of the actuator and the flexures 6 are denoted k a , c a , and k/ , c/, respectively. Accordingly, Newton's second law governs the dynamics of the suspended platform 3, as follows:
• M a is the effective mass of the actuator 4
• M p is the effective mass of the platform 3
• d is the displacement of the platform 3
• Cf is the damping coefficient of the flexures 6; d is the first order derivative; and d is the second order derivative.
• the load force F is of interest, and this can be related to the actuator generated force F ⁇ by applying Newton's second law to the actuator mass, as follows:
• the actuator 4 is a 10 mm long PZT (lead zirconium titanate) linear actuator with 200 layers.
• the force sensing piezoelectric transducer 5 is a single PZT wafer of the same area. The dimensions and physical properties of the system are listed in Table 1 below.
• the full scale displacement is therefore 8.5 ⁇ m at 200 V and the system resonance frequencies are
• the measured load force is also used to damp the resonance frequencies of the nanopositioning system 1 by processing it in the feedback loop with an integral controller, which will be referred to hereinafter as integral force feedback (IFF).
• IFF integral force feedback
• IFF is particularly useful to augment the damping of flexible structures, since it is simple to implement and, under common circumstances, provides excellent damping performance with guaranteed stability.
• An IFF damping controller C d is shown in Figure 10 connected to the system Gy s va, which represents the nanopositioning system 1 having respective actuator and sensor gains g a and g s . Therefore:
• G n M g a ⁇ rr ⁇ g . - (33)
• Gy s va(s) is the system GvsVa, ga i ' s the actuator gain; g s is the sensor gain;
• F a (s) is the force applied by the actuator 4.
• Mp is the effective mass of the platform 3; Mis the sum of the effective masses of the platform 3 and the actuator 4;
• Cf is the damping coefficient of the flexures 6; c is the sum of the damping coefficients of the flexures 6 and the actuator 4; kf is the stiffness of the flexures 6; k is the sum of the stiffnesses of the flexures 6 and the actuator 4; and s is the Laplace transform parameter.
• the system transfer function from the applied voltage to measured voltage Gysva can be derived as,
• the two system transfer functions G d v a and Gysva can be used to simulate the performance of feedback control systems. As both of these transfer functions have the same input V n and poles, it is convenient to define a single-input two-output system G that contains both of these transfer functions,
• phase response lies between 0 and 180 degrees.
• Cd(s) is the integral controller; a is the controller gain; and s is the Laplace transform parameter.
• the integral controller Cd has a constant phase lag of 90 degrees
• the loop-gain phase lies between -90 and 90 degrees. That is, the closed-loop system has an infinite gain-margin and phase-margin of 90 degrees.
• the system Gy sVa has the advantageous properties of simplicity and robustness due to the use of IFF.
• the closed-loop poles are obtained from the roots of the following equation: s(s + ⁇ p 2 )
• the corresponding closed-loop root-locus is plotted in Figure 11, where it can be seen that the closed-loop poles remain in the left half plane, meaning that the system is unconditionally stable.
• the root-locus also provides a simple and convenient method for finding the optimal feedback gain numerically. This is useful where the model parameters are unknown, for example, if the system GvsVa is procured directly from experimental data by system identification.
• the numerically optimal gain is 4.07 x 10 4 which provides a closed-loop damping ratio of 0.45. This correlates closely with the predicted values of equation (41) and supports the accuracy of the assumptions made in deriving the optimal gain.
• the simulated open-loop and closed-loop frequency responses from the disturbance input w to the measured sensor voltage V s in the system 1 employing IFF in accordance with the embodiment of Figure 10 are plotted in the frequency response graphs of Figure 12. In the magnitude graph, the open-loop frequency response is plotted as line 100 while the closed-loop response is plotted as line 101.
• the open-loop frequency response is plotted as line 102 while the closed-loop response is plotted as line 103.
• the measured voltage V s is related to displacement by
• the controller 6 uses the relationship in equation (44) in the feedback loop to calculate the displacement d of the platform 3 from the measured output voltage V s from the transducer 5 in the system 1.
• the measured force is also proportional to the displacement at frequencies below the system zeros.
• the measured load force F via its output voltage V s can be used to calculate the displacement d above or below the response frequency ⁇ .
• the measured load force provides a suitable feedback variable for the controller 6 when positional or trajectory tracking is required within the system.
• V p is the piezoelectric strain voltage
• Ri n is the voltage buffer input impedance
• C is the transducer capacitance
• the filter is high-pass with a cut-off frequency of 1/Ru 1 C.
• the high-pass cut-off frequency can be made extremely low, in the order of 1 mHz, to ameliorate this problem.
• this is not always desirable as the settling time for the system 1 becomes extremely long.
• the system 1 still cannot track DC and the high source impedance results in a noisy measurement due to the buffer input current noise. Therefore, it is preferable to eliminate the cut-off frequency of the piezoelectric sensor by setting a "cross-over" frequency at which the piezoelectric force sensor is not used.
• the inventor proposes to use an auxiliary signal, either an estimate of the actual displacement or a displacement measurement to correct the response at low frequencies.
• the system 1 is modified to incorporate a dual sensor control loop in another embodiment of the invention, as illustrated in Figures 13 to 15, where corresponding features have been given the same reference numerals.
• the nanopositioning stage 150 has a position or displacement sensor in the form of an optical sensor 151 for measuring the position or displacement of the platform 3 when displaced by the actuator 4.
• the optical sensor 151 includes two sensor elements 152 and 153 respectively mounted to the fixed base 2 and the platform 3.
• the optical sensor 151 is electrically connected to the controller 6 via a connection line 154.
• the displacement signals obtained from the optical sensor 151 are transmitted to the controller 6 via a connection line 154 for processing in the feedback loop.
• the nanopositioning stage 150 has a dual sensor feedback loop, comprising the force sensing piezoelectric transducer 5 and the optical displacement sensor 151.
• the optical displacement sensor may be replaced with a capacitive or inductive proximity sensor, if desired.
• the tracking control loop is schematically illustrated in Figure 14, where corresponding features have been given the same reference letters.
• the loop 160 is similar to Figure 8 with the exception of the additional complementary filters F H and F L .
• These complementary filters F H and Fi substitute the displacement measurement d m for the measured load force output voltage V 3 at frequencies below the cross-over frequency ⁇ c , which as discussed above is the frequency, preferably above the cut-off frequency, at which the piezoelectric sensor is deemed to be unreliable due to noise.
• the simplest choices of complementary filters are:
• the cross-over frequency ⁇ c is equal to the bandwidth of the filters F H and Fi.
• displacement sensors are typically noisy over large bandwidths, they have better thermal and drift characteristics than piezoelectric sensors.
• the complementary filters F H and Fi exploit the best aspects of each signal.
• the wide- bandwidth and low noise of piezoelectric force sensors is exploited at frequencies above the cross-over frequency ⁇ c , while the displacement sensors provide a high level of stability at DC and at low frequencies below the cross-over frequency ⁇ c .
• FIG. 15 illustrates the method 170 according to another embodiment of the invention and in which corresponding features have been given the same reference numerals.
• the nanopositioning stage 150 functions in a similar manner as the embodiment of Figure 3. That is, an input signal corresponding to a desired position of the object on the platform 3 is sent to the controller 6, which converts it into a signal corresponding to the necessary displacement du The controller 6 then actuates the piezoelectric actuator 4 through the connection line 7 with the displacement signal. The actuator 4 then applies a force to the platform 3 to displace it a distance d at step 21.
• the piezoelectric transducer 5 measures the load force on the platform 3 at step 22, converts the load force F into an output voltage V 8 and then transmits this output voltage V 3 as a signal corresponding to the measured load force back to the controller 6 through the connection line 8 at step 171.
• the optical displacement sensor 151 measures the displacement d at step 172 using optical sensing elements 152 and 153, converts it into a signal corresponding to the measured displacement d and transmits it back to the controller 6 through the connection line 154 at step 171.
• the controller 6 divides the signal representing the response frequency a>i at step
• the component 174 above ⁇ c is processed by feeding the measured load force signal into the feedback loop and calculating the actual displacement d 2 in accordance with the predetermined proportional relationship of equation (43), as best shown in step 176, as best shown in Figure 14a.
• the component 175 below the cross-over frequency ⁇ c is processed by feeding the measured displacement d into the feedback loop as a substitute for the measured load force at step 177. This is schematically illustrated in Figure 15a.
• the components are then recombined at step 178 to produce the output voltage signal V a from the controller 6 at step 179 that is to be applied to the piezoelectric actuator 4.
• the controller 6 applies the output voltage signal V a through the connection line 7 to the piezoelectric actuator 4 to decrease or increase the force applied to the platform 3, and hence the load force on the platform, thus adjusting the position of the platform 3.
• the stage 1 is able to advantageously dynamically correct the platform position without being adversely affected by sensor- induced noise. This is because at frequencies above the cross-over frequency ⁇ c , the piezoelectric transducer 5 has a wider bandwidth and generates much less noise than standard position sensors, such as inductive and capacitive position sensors.
• the displacement sensor 151 provides a high level of stability and has better thermal and drift characteristics than piezoelectric sensors. In effect, the system 150 and method 170 exploit the best aspects of each signal.
• the controller 6 processes the measured load force in the feedback loop to increase the damping ratio ⁇ for the stage 150. As discussed above, this proportionately increases the feedback gain and the closed-loop bandwidth. Thus, the stage 150 is able to operate in a larger closed-loop bandwidth while meeting the condition for closed-loop stability in accordance with equation (2).
• the displacement signal d is obtained by using a physical displacement sensor 150
• the displacement signal d can instead be derived by an estimate calculated from the input voltage V a and the open-loop response of the system. This may be suitable where a physical displacement sensor is not available or the system does not require a high level of accuracy at DC.
• Figure 16a A further embodiment is illustrated in Figure 16, which the inventor has called a low frequency bypass employing an estimated displacement signal described above, and where corresponding features have been given the same reference numerals.
• the loop 180 is similar to the loop 160 of Figure 14a, with the exception that the complementary filters F H and F L substitute an estimated displacement d for the measured load force output voltage V 3 at frequencies below the cross-over frequency ⁇ c .
• the measured load force is still used for tracking control for frequencies above the cross-over frequency ⁇ c and/or damping control, as in Figure 14, but no displacement measurement or sensor is used.
• the signal F 0 requires the same sensitivity as V 8 , so the scaling constant ⁇ is
• a feedforward input % can be optionally used to improve the closed-loop response of the system, as best shown in Figures 14a and 16a.
• Inversion based feedforward provides the best performance but the additional complexity is undesirable for analog implementation. Accordingly, a basic but effective form of feedforward compensation is to simply use the inverse DC gain of the system as a feedforward injection filter, i.e.
• the line 200 is the frequency response of the basic controller
• the line 201 is the frequency response of the direct tracking controller
• line 202 is the controller C with IFF (using a dual sensor loop or a low frequency bypass).
• IFF using a dual sensor loop or a low frequency bypass
• the magnitude response graph indicates that the controllers employing force feedback (direct tracking or IFF) in accordance with the embodiments of Figures 5 and 13 to 16 provide a bandwidth close to the resonance frequency of the system, which corresponds to the open-loop frequency response, due to the high gain of the feedback loop.
• the controllers employing force feedback provide exceptional performance, especially when considering the large stability margins and controller simplicity.
• the closed-loop frequency response of the basic controller, the direct tracking controller, the integral controller using IFF with a dual sensor or a low frequency bypass are compared using a 500 Hz triangle reference wave, which is illustrated by the line 300.
• the line 301 is the open-loop response of the system 1 without feedback
• the line 302 is the frequency response of the integral controller using IFF with a dual sensor or low-frequency bypass
• the line 303 is the frequency response of the direct tracking controller
• the line 304 is the frequency response of the basic integral controller. It can be seen from the graph that the frequency of operation is beyond the capacity of the basic integral controller, as indicated by line 304.
• the controller implementing IFF with a dual sensor or low-frequency bypass (line 302) provides sufficient bandwidth to achieve good tracking performance, even with an input frequency only 1 decade below the system resonance, as shown by the relatively close proximity of the line 302 to the reference line 300 and the open-loop line 301.
• the direct tracking controller using the measured load force for damping control (line 303) still provided adequate bandwidth for improved tracking performance, but had greater tracking lag.
• the examples using the embodiment of the invention incorporating a dual sensor feedback loop using force and displacement as the feedback variables demonstrate the effectiveness of the proposed tracking and damping controller implementing the method and system of the invention.
• this embodiment overcomes the high -pass characteristic exhibited by piezoelectric sensors at low frequencies by replacing the low frequency force signal with a displacement measurement or displacement signal estimated from the open-loop system dynamics.
• the dual sensor/low frequency bypass integral force feedback controller in this embodiment of the invention provides a closed-loop bandwidth approaching the open-loop resonance frequency while maintaining an infinite gain margin and 90° phase margin.
• a basic integral displacement feedback controller achieves only 1% of the bandwidth with a gain-margin of only 5 dB.
• FIG. 19a An example in accordance with the embodiments of the invention will now be described with reference to Figures 19 and 20.
• the dual sensor loop configuration of Figure 15 was applied to a high-bandwidth lateral nanopositioning platform designed for video speed scanning probe microscopy, as best shown in Figure 19a.
• This device is a serial kinematic device with two moving stages both suspended by leaf flexures and driven directly by 10mm stack actuators. A small stage is located in the centre and is designed for scan-rates up to 5 kHz so as to be sufficiently fast with a resonance frequency of 29 kHz. A larger stage provides motion in the adjacent axis and is limited by a resonance frequency of 1.5 IcHz. As this stage is required to operate with triangular trajectories up to 100 Hz, active control is required.
• the main application for this nanopositioning platform is high speed scanning probe microscopy, where high resolution and wide bandwidth are the most desirable performance characteristics.
• the platform is mechanically similar to the system in Figure 13. The major difference is the existence of higher frequency modes beyond the first resonance frequency. These can be observed in the open-loop frequency response plotted in Figure 20a. While only a single mode system has been discussed for simplicity, it will be appreciated that the method and system of the invention is readily applicable to higher order resonance modes. It has been found that IFF is especially suited to resonant systems of high-order. This is due to the large feedthrough term in equation (25), which guarantees zero-pole ordering regardless of system order. Hence, the excellent stability characteristics are not affected by the existence of higher-order resonance modes.
• the nanopositioning platform has two sensors installed, an ADE Tech 2804 capacitive sensor and a piezoelectric force sensor.
• piezoelectric force sensor Two types were tested, a standard plate sensor, as best shown in Figure 19b, and a custom stack actuator with an integrated force sensor, as best shown in Figure 19c.
• the plate sensor is only a single layer of piezoelectric material with metal electrodes on the top and bottom faces.
• the stack actuator is 5> ⁇ 5 ⁇ 10 mm in dimensions and was manufactured by Noliac A/S, Denmark. This transducer is mechanically stronger than the plate sensor and due its larger capacitance, is less sensitive to current noise.
• the actuator was driven with a high- voltage amplifier of the inventor's own design.
• the closed-loop frequency response is plotted in Figure 20b and reveals significant damping of the first three modes by 24, 9 and 4 dB.
• the simulated response is also overlain which shows a close correlation.
• the tracking bandwidth of the closed-loop system is 2.07 kHz, which is higher than the open-loop resonance frequency and significantly greater than the bandwidth achievable with a direct tracking controller, predicted to be 210 Hz with a 5 dB gain-margin.
• Another major benefit associated with the piezoelectric force sensor is its extremely low additive noise.
• the embodiments of the invention add a force sensor into a nanopositioning system to measure the load force applied to the platform and thus enable the controller to use the measured load force to control the position of the platform and/or damp the resonant frequencies of the system.
• the measured load force is used to calculate the platform displacement.
• a piezoelectric sensor can be used as the force sensor, and so generates a lesser amount of sensor-induced noise compared to existing displacement sensors.
• the resulting transfer function from applied voltage to measured load force exhibits a zero-pole ordering which greatly simplifies the design and implementation of the damping controller, permitting exceptional damping performance to be achieved with a simple integral controller without any limitations on the gain.
• the system according to the invention is thus stable with a theoretically infinite gain margin and 90 degrees phase margin. Other outstanding characteristics include guaranteed stability and insensitivity to changes in resonance frequency.
• the increased bandwidth and resolution offered by the invention allows nanopositioning systems implementing the invention to be employed in a new range of high-speed applications.
• high-speed scanning probe microscopes currently use open-loop nanopositioning devices.
• Due to the simplicity and bandwidth of the invention, such applications can now utilise closed-loop control with the associated benefits of improved linearity, less vibration and rejection of disturbance.
• the invention represents a practically and commercially significant improvement over the prior art.

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